If $AX = B$, the matrices $A$ and $B$ are known, and $A$ is not a square matrix. It is desired to solve for the value of the matrix $X$. The elements of the matrix are 0/1, i.e., they are over a binary field $\mathbb F_2$.
I only know how equations on the real number field should be solved. I hope I can get help from you, thanks a lot!
Let $x_k$ and $b_k$ be the $k$-th column of $X$ and $B$ respectively. Then $Ax_k=b_k$. This linear system can be solved with Gaussian elimination over every field.